By Chern S.S., Li P., Cheng S.Y., Tian G. (eds.)

Those chosen papers of S.S. Chern talk about subject matters similar to critical geometry in Klein areas, a theorem on orientable surfaces in 4-dimensional house, and transgression in linked bundles

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A simplicia1 subdivision L,, of L, and an almost semilinear realization HrL,,O=L:,O of L, in R" through the subdivision L,,, such that the following conditions are satisfied: 1". L,,,=K,x ( l ) + K , x ( 2 ) + C 6 ; X [l, 21. iC1, 2". H,(T~x ( j ) ) = T j ( T j ) ,r j C K j , j=1, 2 and for ;Elo, R(Zix [ l , 2 ] ) is a simple broken line li. 3". o = H,-I. 4". x [ 1,2] n Rr(Zjx [ 1,2])=@. For the construction let us first draw in R" for each iC], a simple broken line 1; joining x ( i ) ) ,j = 1, 2, such that these l; together with T,K,+T,K, form an almost euclidean complex.

The projection by A: R(b x r) = b * r Then by the definition of Hence we have Theorem 6. *). qr we have evidently be the covering projection of = $=, n* #a R* @" = p, ( m even 6"', Denote ( m odd) > 0) A ' pT = ,qm) qT. g" on K", then (17) (18) in which pz denotes reduction mod 2. Suppose for the moment 6"'= 0 (cf. the remark above), then (17) and (18) may be reduced simply to x*@"'=O, 37 m>O. (19) 266 $ 4 . THEREALIZABILITY OF ANY COCYCLE IN THE IMBEDDING CLASSES W e have proved that the m-dimensional imbedding cochains FT (or pT) of K are all cocycles and belong to one and the same cohomology class 6'"E g"''* ( K ) (or 0"' E H"" ( K , Zcm)).

21 This page intentionally left blank SCIENTIA SINICA Vol. VII, No. 3, 1958 MATHEMATICS ON THE REALIZATION OF COMPLEXES IN EUCLIDEAN SPACES I" ABSTRACT I t was early known that a n y n-dimensional abstract complex may be realized in a ( 2 n + 1)-dimensional euclidean space RZnf'. From this theorem, whose proof is quite simple, i t follows that the ( 2 n + l ) -dimensional euclidean space contains in reality all imaginable n-dimensional complexes. 2+1, is a problem much more difficult which cannot, i t seems, be solved completely i n the near future.